Note for the Prime Gaps

A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted gₙ is the difference between the (n + 1)st and the nth prime numbers, i.e. gₙ=pₙ₊₁-pₙ. There isn't a verified solution to Andrica's conjecture yet. The conjecture itself deals with the difference between the square roots of consecutive prime numbers. While mathematicians have showed it true for a vast number of primes, a general solution remains elusive. We consider the inequality θ(pₙ₊₁)θ(pₙ) ≥ √ pₙ₊₁pₙ for two successive prime numbers pₙ and pₙ₊₁, where θ(x) is the Chebyshev function. In this note, under the assumption that the inequality θ(pₙ₊₁)θ(pₙ) ≥ √ pₙ₊₁pₙ holds for all n ≥ 1.3002 · 10¹⁶, we prove that the Andrica's conjecture is true. Since θ(pₙ₊₁)θ(pₙ) ≥ √ pₙ₊₁pₙ holds indeed for large enough prime number pₙ, then we show that the statement of the Andrica's conjecture can always be true for all primes greater than some threshold.